Bernstein's problem

In differential geometry, Bernstein's problem is as follows: if the graph of a function on Rⁿ⁻¹ is a minimal surface in Rⁿ, does this imply that the function is linear? This is true for n at most 8, but false for n at least 9. The problem is named for Sergei Natanovich Bernstein who solved the case n = 3 in 1914.

Statement

Suppose that f is a function of n − 1 real variables. The graph of f is a surface in Rⁿ, and the condition that this is a minimal surface is that f satisfies the minimal surface equation

$\sum _{i=1}^{n-1}\frac{\partial }{\partial x_i}\frac{\frac{\partial f}{\partial x_i}}{\sqrt{1+\sum _{j=1}^{n-1}{\left(\frac{\partial f}{\partial x_j}\right)}^2}}=0$

Bernstein's problem asks whether an entire function (a function defined throughout Rⁿ⁻¹ ) that solves this equation is necessarily a degree-1 polynomial.

History

proved Bernstein's theorem that a graph of a real function on R² that is also a minimal surface in R³ must be a plane.

gave a new proof of Bernstein's theorem by deducing it from the fact that there is no non-planar area-minimizing cone in R³.

showed that if there is no non-planar area-minimizing cone in Rⁿ⁻¹ then the analogue of Bernstein's theorem is true for graphs in Rⁿ, which in particular implies that it is true in R⁴.

showed there are no non-planar minimizing cones in R⁴, thus extending Bernstein's theorem to R⁵.

showed there are no non-planar minimizing cones in R⁷, thus extending Bernstein's theorem to R⁸. He also showed that the surface defined by

$\{x\in {ℝ}^8:x_1^2+x_2^2+x_3^2+x_4^2=x_5^2+x_6^2+x_7^2+x_8^2\}$

is a locally stable cone in R⁸, and asked if it is globally area-minimizing.

showed that Simons' cone is indeed globally minimizing, and that in Rⁿ for n≥9 there are graphs that are minimal, but not hyperplanes. Combined with the result of Simons, this shows that the analogue of Bernstein's theorem is true in Rⁿ for n≤8, and false in higher dimensions.

See also

• Simons cone

References

• Almgren, F. J. (1966), "Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem", Annals of Mathematics, Second Series, 84 (2): 277–292, doi:10.2307/1970520, ISSN 0003-486X, JSTOR 1970520, MR 0200816
• Bernstein, S. N. (1915–1917), "Sur une théorème de géometrie et ses applications aux équations dérivées partielles du type elliptique", Comm. Soc. Math. Kharkov, 15: 38–45 German translation in Bernstein, Serge (1927), "Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus", Mathematische Zeitschrift (in German), 26, Springer Berlin / Heidelberg: 551–558, doi:10.1007/BF01475472, ISSN 0025-5874{{citation}}: CS1 maint: unrecognized language (link)
• Bombieri, Enrico; De Giorgi, Ennio; Giusti, E. (1969), "Minimal cones and the Bernstein problem", Inventiones Mathematicae, 7 (3): 243–268, Bibcode:1969InMat...7..243B, doi:10.1007/BF01404309, ISSN 0020-9910, MR 0250205, S2CID 59816096
• De Giorgi, Ennio (1965), "Una estensione del teorema di Bernstein", Ann. Scuola Norm. Sup. Pisa (3), 19: 79–85, MR 0178385
• Fleming, Wendell H. (1962), "On the oriented Plateau problem", Rendiconti del Circolo Matematico di Palermo. Serie II, 11: 69–90, doi:10.1007/BF02849427, ISSN 0009-725X, MR 0157263
• Simons, James (1968), "Minimal varieties in riemannian manifolds", Annals of Mathematics, Second Series, 88 (1): 62–105, doi:10.2307/1970556, ISSN 0003-486X, JSTOR 1970556, MR 0233295

External links

• Encyclopaedia of Mathematics article on the Bernstein theorem